Which of the following statements (s) is/are TRUE ?
I. `sqrt(12) gt root3(16) gt root4(24)`
`II. root3(25) gt root4(32) gt root6(48)`
III. `root4(9) gt root3(15) gt root6(24)`

Options

A. Only I and II
B. Only I and III
C. Only I
D. All are true

Answer

To determine which of the given statements is true, we will evaluate each statement step by step.

### Statement I: \( \sqrt{12} > \sqrt[3]{16} > \sqrt[4]{24} \)

1. **Calculate \( \sqrt{12} \)**:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \approx 3.464 \]

2. **Calculate \( \sqrt[3]{16} \)**:
\[ \sqrt[3]{16} = 16^{1/3} = (2^4)^{1/3} = 2^{4/3} \approx 2.52 \]

3. **Calculate \( \sqrt[4]{24} \)**:
\[ \sqrt[4]{24} = 24^{1/4} = (2^3 \times 3)^{1/4} = 2^{3/4} \times 3^{1/4} \approx 2^{0.75} \times 1.316 \approx 2.66 \]

4. **Compare the values**:
\[ 2\sqrt{3} \approx 3.464 > 2.52 > 2.66 \]
This statement is **TRUE**.

### Statement II: \( \sqrt[3]{25} > \sqrt[4]{32} > \sqrt[6]{48} \)

1. **Calculate \( \sqrt[3]{25} \)**:
\[ \sqrt[3]{25} = 25^{1/3} = (5^2)^{1/3} = 5^{2/3} \approx 3.684 \]

2. **Calculate \( \sqrt[4]{32} \)**:
\[ \sqrt[4]{32} = 32^{1/4} = (2^5)^{1/4} = 2^{5/4} \approx 2.828 \]

3. **Calculate \( \sqrt[6]{48} \)**:
\[ \sqrt[6]{48} = 48^{1/6} = (2^4 \times 3)^{1/6} = 2^{2/3} \times 3^{1/6} \approx 2^{0.6667} \times 1.2009 \approx 2.0 \]

4. **Compare the values**:
\[ 3.684 > 2.828 > 2.0 \]
This statement is **TRUE**.

### Statement III: \( \sqrt[4]{9} > \sqrt[3]{15} > \sqrt[6]{24} \)

1. **Calculate \( \sqrt[4]{9} \)**:
\[ \sqrt[4]{9} = 9^{1/4} = (3^2)^{1/4} = 3^{1/2} = \sqrt{3} \approx 1.732 \]

2. **Calculate \( \sqrt[3]{15} \)**:
\[ \sqrt[3]{15} = 15^{1/3} \approx 2.466 \]

3. **Calculate \( \sqrt[6]{24} \)**:
\[ \sqrt[6]{24} = 24^{1/6} \approx 2.0 \]

4. **Compare the values**:
\[ 1.732 < 2.466 > 2.0 \]
This statement is **FALSE**.

### Conclusion:
The true statements are **I and II**.